172 
GENETICS: STURTEVANT, BRIDGES, AND MORGAN 
the line is curved instead of straight. This curvature is due to the phenome- 
non of double crossing over, and the degree of curvature is dependent on the 
frequency of double crossovers; i.e., upon 'coincidence.' This method of 
representing coincidence, however, leads to inconsistencies. As was pointed 
out above, it is possible to build up the whole X-chromosome from succes- 
sive overlapping sections, each of which is so short as to include few or no 
double crossovers. Yet double crossovers are not rare in longer sections. If 
ABC and BCD are sections so short that no double crossing over occurs 
within each, section ABCD may be long enough so that double crossing over 
in which one crossover is between A and B and the other between C and D 
may still occur; and in that case it becomes necessary to represent D in two 
positions at the same time. Such situations are actually known in Droso- 
phila. Hence the curved line cannot give a consistent scheme; nor can any 
other scheme based on the assumption that long distances, as well as short 
ones, are to be represented as proportional to observed percentages of crossing 
over. 
According to Castle the supposition that double crossovers do occur is "an 
unproved secondary hypothesis." Of course, if three genes are imagined as 
not lying in a straight line, a single plane may separate any one from the 
other two. Under these conditions double crossing over has no meaning. 
But the fact remains that when three linked genes are studied simultaneously, 
one pair of contrary classes is always small, and this class is always the one 
that is the double crossover class on the linear arrangement scheme. Some 
method of accounting for the smallness of this class is evidently demanded. 
Castle recognizes this, and suggests that the explanation may be either that 
"only transverse breaks occur, of which two taking place simultaneously are 
required to produce the difficult regrouping" (i.e., the familiar double crossover 
explanation), or that "transverse breaks are more frequent than oblique longi- 
tudinal ones, of which a single one would suffice to accomplish the regrouping, 
if the genes are not strictly linear in arrangement." MuUer^ has published 
data bearing on this point. He followed simultaneously eleven loci in the 
X-chromosome of Drosophila, and obtained some double crossover classes that 
according to Castle's model are impossible with a single plane. However, if 
the loci all reduce to the curved line discussed above, than any so-called double 
crossover is possible with a single separation plane. But those classes now 
termed triple and quadruple crossovers are all impossible with a single plane of 
crossing over. In the case of the X-chromosome over a dozen triple crossovers 
have been observed, and in the case of the second chromosome they are so 
frequent that 131 were observed in a single experiment. It follows that, 
even if the genes are not arranged in a straight line, the occurrence of double 
crossing over is an established fact, not an unproved hypothesis. 
If crossover planes do not occur longitudinally, or occur thus less often than 
transversely, it is difficult to see how the distances apart of the loci in Castle's 
model can be proportional to the crossover values, except in the case where 
